The continuum mechanical theory of multicomponent diffusion in fluid mixtures

Chemical Engineering Science 65 (2010) 5976–5989

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The continuum mechanical theory of multicomponent diffusion in fluid mixtures Ravindra Datta n, Saurabh A. Vilekar Fuel Cell Center, Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609, United States

a r t i c l e in f o

a b s t r a c t

Article history: Received 15 September 2009 Received in revised form 16 August 2010 Accepted 20 August 2010 Available online 27 August 2010

The continuum mechanical approach for deriving the generalized equations of multicomponent diffusion in fluids is described here in detail, which is based on application of the principle of linear momentum balance to a species in a mixture, resulting in the complete set of diffusion driving forces. When combined with the usual constitutive equations including the continuum friction treatment of diffusion, the result is a very complete and clear exposition of multicomponent diffusion that unifies previous work and includes all of the various possible driving forces as well as the generalized Maxwell–Stefan form of the constitutive equations, with reciprocal diffusion coefficients resulting from Newton’s third law applied to individual molecular encounters. This intuitively appealing and rigorous approach, first proposed over 50 years ago, has been virtually ignored in the chemical engineering literature, although it has a considerable following in the mechanical engineering literature, where the focus, naturally, has been physical properties of multiphase fluid and solid mixtures. The described approach has the advantages of transparency over the conventional approach of non-equilibrium thermodynamics and of simplicity over those based on statistical mechanical or kinetic theory of gases or liquids. We provide the general derivation along with some new results in order to call attention of chemical engineers to this comprehensive, attractive, and accessible theory of multicomponent diffusion in fluids. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Multicomponent diffusion Maxwell–Stefan equation Mixture theory Thermal diffusion Stress diffusion Pressure diffusion

1. Introduction The theory of multicomponent diffusion and its various driving forces is treated in a large number of chemical engineering books and articles (of necessity, only representative publications are cited here), e.g., for: (1) fluids of different state of aggregation, i.e., in gaseous, liquid, electrolyte, polymer, and colloidal solutions (Bird et al., 2002; Curtiss and Bird, 1996, 1999; Cussler, 1976, 1997; Deen, 1998; Kerkhof and Geboers, 2005a, 2005b; Kuiken, 1994; Lightfoot, 1974; Newman, 1991; Slattery, 1981, 1999; Taylor and Krishna, 1993; Tyrrell and Harris, 1984; Wesselingh and Krishna, 1990); (2) porous media (Do, 1998; Jackson, 1977; Mason and Malinauskas, 1983; Mason et al., 1967; Whitaker, 1986, 1999); and (3) membranes (Datta et al., 1992; Mason and Lonsdale, 1990; Spiegler, 1958; Thampan et al., 2000). Although this general multicomponent diffusion theory is being increasingly utilized in the rigorous analysis of many chemical engineering mass transport problems (Amundson et al., 2003; Bird et al., 2002; Cussler, 1997; Datta and Rinker, 1985; Jackson, 1977; Krishna, 1987c; Krishna and Wesselingh, 1997; Lightfoot,

n

Corresponding author. E-mail address: [email protected] (R. Datta).

0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.08.022

1974; Taylor and Krishna, 1993; Thampan et al., 2000; Wang and Datta, 1986), starting with the early work of Toor (1957) and Stewart and Prober (1964), its application is still somewhat limited. We feel that a reason for this is the lack of a clear exposition of the underlying principles of the multicomponent diffusion theory. The conventional derivation of multicomponent diffusion in continua based on the non-equilibrium, or irreversible, thermodynamics, IRT (Bird et al., 2002; Curtiss and Bird, 1999; de Groot and Mazur, 1962) can be mystifying (Cussler, 1997), while that based on statistical mechanical or kinetic theory of gases (Hirschfelder et al., 1964) or liquids (Bearman and Kirkwood, 1958; Snell et al., 1967) can be intimidating and, thus, not generally discussed at length in the chemical engineering undergraduate or graduate curricula. As a result, the origin and the basis of the rather complex equations of multicomponent diffusion remain obscure to most chemical engineers. It is our purpose here, therefore, to provide their derivation based on the more transparent approach of species linear momentum conservation (Snell and Spangler, 1967; Truesdell, 1957, 1962), involving a balance of diffusion ‘‘drag’’ and diffusion ‘‘driving forces,’’ and grounded in the familiar principles of the subject of continuum Transport Phenomena (Bird et al., 2002). Also provided are some clarifications and new results. The derivation provides very

R. Datta, S.A. Vilekar / Chemical Engineering Science 65 (2010) 5976–5989

general results and without some of the approximations inherent in the other approaches. The diffusive flux equations in multicomponent mixtures are typically written in two alternate forms, i.e., either in: (1) the generalized Fick–Onsager (GFO) form (Onsager, 1945), a generalization of Fick’s law (Fick, 1855), in which the diffusive flux of a species i is written as a linear combination of the diffusion ‘‘driving forces’’ dj for all species (Fitts, 1962) or in (2) the generalized Maxwell–Stefan form (GMS) (Lightfoot et al., 1962; Maxwell, 1866, 1868; Stefan, 1871) in which the diffusion driving force for the ith species di is written as a linear combination of the fluxes of all species. Owing to the assumed linearity between flux and driving forces, the GFO and the GMS forms are, in fact, interconvertible through the use of linear algebra (Bearman, 1959; Fitts, 1962). Alternatively, the continuum frictional approach (Laity, 1959; Lamm, 1957; Spiegler, 1958) provides expressions in the form of the GMS equations but involves frictional, or impedance, coefficients, zij, rather than the GMS binary diffusion coefficients, Dij. The generalized diffusion driving force for the ith species di comprises not only the usual composition gradient, but also thermal and pressure gradients, stresses, as well as external body forces. The usual approach to obtaining these different driving forces is that of non-equilibrium, or irreversible thermodynamics, i.e., IRT (de Groot, 1951; de Groot and Mazur, 1962; Eu, 1992; Fitts, 1962; Haase, 1969; Kuiken, 1994; Lightfoot, 1974; Merk, 1959; Yao, 1981); starting with the entropy balance (Jaumann, 1911), and incorporating in it the equations of mass, momentum, and energy balance. An inspection of the resulting entropy production term shows it to be a sum of the products of fluxes and driving forces, thus identifying the various diffusion driving forces contained in di. The IRT approach furthermore utilizes the linearity postulate, i.e., that the flux of a species involves a linear combination of the driving forces of all species dj (Bird et al., 2002). The results are applicable to any mixture, although the transport coefficients are treated as phenomenological coefficients satisfying certain symmetry, i.e., the Onsager reciprocal relations (ORR) (Monroe and Newman, 2006), and other constraints. A criticism of the IRT theory is provided by Truesdell and others (Truesdell, 1984), e.g., its assertion of a lack of coupling between mass and viscous dissipation fluxes and forces, owing to their different tensorial rank based on the postulate of Curie, and the seemingly arbitrary choice for the reference frame for flux. The equations of multicomponent diffusion for gases, on the other hand, are derived based on the Chapman–Enskog approach (Chapman and Cowling, 1970; Eu, 1992; Ferziger and Kaper, 1972; Hirschfelder et al., 1964) to the solution of the Boltzmann equation, involving the assumption of linear deviations from the equilibrium Maxwell–Boltzmann distribution function for component i, in the form of a perturbation function, that provides both the different diffusion driving forces, i.e., concentration and pressure gradient and external forces, as well as the Maxwell– Stefan constitutive equation along with predictive expressions for the diffusion coefficients involved. The alternate approach for gases is the Grad–Zhdanov theory (Grad, 1949; Zhdanov et al., 1962), in which the distribution function for component i is written as a product of the equilibrium distribution function and a series of Hermite polynomials, and moments of the Boltzmann equation are generated for species i. The final result is modified expressions for the diffusion coefficients and the appearance of a stress term in the diffusion driving force that is absent in the Chapman–Enskog theory as well as in IRT. The Grad–Zhdanov theory also forms the starting point for the derivation of the Dusty-Gas Model (DGM) for gaseous transport in porous media (Cunningham and Williams, 1980; Mason et al., 1967; Kerkhof, 1996; Weber and Newman, 2005). The DGM has, unfortunately, also so far seen rather limited use in chemical engineering

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applications (Abed and Rinker, 1973; Datta et al., 1992; Datta and Rinker, 1985; Kaza and Jackson, 1980; Krishna, 1987b; Skrzypek et al., 1984; Suwanwarangkula et al., 2003; Thampan et al., 2001; Wang and Datta, 1986). For the case of monatomic liquids, Bearman and Kirkwood (1958) derived the complete multicomponent diffusion equations in the form of partial momentum balance of species i from statistical mechanics of molecular dynamics based on the Liouville equation. They determined the perturbation in distribution functions due to departure from equilibrium and provided formal integral equations for transport coefficients. The Bearman– Kirkwood theory forms the starting point for the derivation of the Dusty-Fluid Model (DFM) for liquid transport in porous media and in membranes (Mason and Lonsdale, 1990; Mason and Viehland, 1978). The DFM has also seen only rather limited application so far (e.g., Noordman et al., 2002; Thampan et al., 2000; Syed and Datta, 2002). Curtiss and Bird (1996) have extended the Bearman– Kirkwood theory to polymeric liquids. An alternate approach that has received scant attention in the chemical engineering literature, with few notable exceptions (Curtiss and Bird, 1996; Whitaker, 1986), although it is wellknown in the mechanical engineering literature as the ‘‘Theory of Mixtures’’ (Bowen, 1976; Green and Naghdi, 1967) and applied to multiphase fluid and solid mixtures, is the theory of diffusion based on the linear momentum balance of species (Atkin and ¨ Craine, 1976b; Bowen, 1967; Muller, 1968; Snell and Spangler, 1967; Stefan, 1871; Truesdell, 1957, 1962, 1984; Truesdell and Toupin, 1960; Williams, 1958). This is unfortunate, since it provides a remarkably complete description of the various diffusion driving forces based on more readily understood continuum mechanical arguments. After all, diffusion involves relative ‘‘motion’’ of a species in response to driving ‘‘forces.’’ It, therefore, stands to reason that a reliable description of diffusion should result from species linear momentum balance. When combined with the ‘‘frictional drag’’ model, another wellaccepted continuum mechanical model, it provides an inherently consistent and detailed description of multicomponent diffusion in fluid mixtures, involving both the various driving forces as well as the constitutive equations of the Maxwell–Stefan form. Of course, the transport coefficients are not predicted accurately by this elementary approach, for which one must resort to molecular theories (Reid et al., 1987) or experiments, although the right forms of expressions result (Furry, 1948; Maxwell, 1868; Ramshaw, 1993; Williams, 1958). Since the molecular theories (Hirschfelder et al., 1964; Kerkhof and Geboers, 2005a; Snell et al., 1967; Zhdanov et al., 1962) differ primarily in this regard, they are complementary, rather than competitors, to the continuum mechanical theory. Additionally, the constraints on the constitutive equations imposed by the Clausius–Duhem entropy inequality have been discussed at length ˇ ´, in the mechanical engineering literature (Samohy´l and Silhavy 1990). A limitation of a purely mechanical theory such as this is, of course, its inability to predict chemical mechanisms of diffusion. Thus, a notable exception is the Grotthuss mechanism (Choi et al., 2005; Grotthuss, 1806) for the anomalous proton diffusion, which, incidentally, predates Fick’s law (Fick, 1855) by half a century! The purpose of this paper, thus, is to provide a complete and self-contained derivation of the multicomponent diffusion equations based on the principle of linear species momentum balance along with the application of Newton’s third law in an elementary treatment of frictional encounters among constituent molecules. This approach provides the most complete form of multicomponent diffusion equations available in the literature, derived normally via more esoteric means inaccessible to the nonspecialist, along with some additional new details. The simplicity of the approach and the underlying principle would hopefully encourage its further usage in chemical engineering education and research.

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second is the non-equilibrium, non-isotropic stress tensor, s. Thus, the divergence of the molecular stress for the mixture

2. Theory 2.1. Linear species momentum balance

r Up ¼ Although the results of this subsection are available in the literature (Atkin and Craine, 1976b; Bowen, 1967; Jakobsen, 2008; ¨ Muller, 1968; Snell and Spangler, 1967; Stefan, 1871; Truesdell, 1957, 1962, 1984; Truesdell and Toupin, 1960; Williams, 1958), we provide the complete derivation here so that the treatment is self-contained. We start by applying the principle of conservation of linear momentum of a component i (i¼1, 2 ,y, n), i.e., of the quantity rivi, to a control volume Vi(t), with a bounding surface Si(t), and translating with the continuum velocity of the species i, vi(r,t) at a position r and time t, within a fluid-phase multicomponent mixture comprising n species, i.e., Z Z XZ d ri vi dV ¼ pi Un dSþ ri f ia dV dt Vi ðtÞ Si ðtÞ a Vi ðtÞ Z Z þ ci Fi dV þ ðMi Dri Þvir dV ð1Þ Vi ðtÞ

Vi ðtÞ

ð2Þ

¨ which was first obtained by Truesdell (Bowen, 1967; Muller, 1968; Snell and Spangler, 1967; Stefan, 1871; Truesdell, 1957, 1962, 1984; Truesdell and Toupin, 1960; Williams, 1958) from species momentum balance, and by Bearman and Kirkwood (1958) from statistical mechanics of molecular dynamics based on the Liouville equation, but in the absence of reaction. In Eq. (2), pi is the partial molecular stress tensor associated with component i (Bearman and Kirkwood, 1958; Snell and Spangler, 1967), so that the total molecular stress tensor for the mixture

p¼

n X

pi ¼ pd þ s

rUpi ¼ rp þ rUs

ð3Þ

ð4Þ

i¼1

P We next assume in Eq. (3) the relation s ¼ ni¼ 1 si for the partial non-equilibrium stress also. Further, from the Gibbs– Duhem equation applied to an isothermal system under mechanP ical and local thermodynamic equilibrium, rp ¼ ni¼ 1 ci rT mi (Fitts, 1962; Horne, 1966), where mi is the chemical potential. Thus, the divergence of the partial molecular stress tensor term in Eq. (2) may be written as

rUpi ¼ ci rT mi þ rUsi

ð5Þ

Using this on the right-hand side of Eq. (2), and further expanding the terms on its left-hand side and using in it the equation of continuity for species i (Bird et al., 2002), i.e., @ri þ rUri vi ¼ Mi Dri @t

on the right-hand side of which, the first term represents the surface forces acting on the component i within the control volume, where pi is the partial molecular stress tensor associated with i, the second term involves external body forces of type a (i.e., gravity, centrifugal, electrostatic, magnetic, etc.), acting on unit mass of i, fia, the third term involves Fi , the internal force of interaction exerted on each mole of i by other constituents within the control volume. The last term in Eq. (1) represents the rate of production of momentum of i by virtue of the net rate of production of the P species i, Dri, from q chemical reactions, i.e., Dri ¼ qr ¼ 1 nri rr , where rr is the molar rate of reaction r, nri is the stoichiometric coefficient of species i in reaction r, and Mi is its molar mass. It is further assumed that the bulk velocity of the produced molecules vir is the same as the bulk velocity of the species, i.e., vir ¼vi, based on local thermodynamic equilibrium. Of course, vir need not, in general, be equal to vi (Jakobsen, 2008). The continuum velocity of a component vi is, of course, that averaged over its molecular velocity distribution, and is related to the massP averaged continuum velocity of the mixture by rv ¼ ni¼ 1 ri vi . With the application of the general transport theorem to the term on the left-hand side of Eq. (1), and the use of Gauss’ divergence theorem for the surface integral term in its right-hand side, all terms in Eq. (1) are rendered as volume integrals and can, consequently, be collected as integrand of a single integral. Furthermore, since the control volume is chosen arbitrarily, the equation must hold for the integrand as well, resulting in the continuum species momentum balance X @ðri vi Þ ri f ia þci Fi þ ðMi Dri Þvi þ rUri vi vi ¼ rUpi þ @t a

n X

ð6Þ

results in (Snell and Spangler, 1967; Truesdell, 1962)   X Dv @vi þ vi Urvi ¼ ci rT mi þ rUsi þ ri ai  ri i i ¼ ri ri f ia þ ci Fi Dt @t a ð7Þ where ai is the partial acceleration, i.e., that of species i (Snell and Spangler, 1967), and Di/Dt  q/qt+ viUr (where r is the del operator) is the substantial derivative for species i (Atkin and Craine, 1976b) which follows the motion of the constituent i, rather than D/Dt ¼q/qt +vUr, which follows the mean motion of the mixture (Bird et al., 2002). It is noteworthy that, unlike in Eq. (2), the reaction term is absent in Eq. (7) because of the use in it of equation of continuity, Eq. (6). When this constituent momentum balance is summed over all species (i¼1, 2, y, n), a momentum balance equation for the mixture is obtained that is of the conventional form, i.e., of the form of linear momentum balance of a single component system (Bird et al., 2002), with the exception of a term involving the summation of riuiui over all species, where ui  vi–v is the diffusion velocity with respect to the mass average velocity of the mixture, v. Therefore, it is expedient to define a total partial stress tensor, sti  si þ sD i , that includes both the partial molecular stress tensor si, termed ‘‘molecular force’’ contribution by Bearman and Kirkwood (1958), as well as the partial diffusive stress tensor, sDi , i.e., the dyad due to the diffusion velocities sDi ¼ ri ui ui , the apparent stress arising from diffusion and termed ‘‘kinetic’’ contribution by Bearman and Kirkwood (1958), and reminiscent of the Reynold’s stresses in turbulent flow. In other words, the total partial non-equilibrium stress (Atkin and Craine, 1976b)

sti ¼ si ri ui ui

ð8Þ

With this, the species momentum balance, Eq. (7), may be rewritten in the form X Dv ri ri f ia þci Fi ð9Þ þ ei ¼ ci rT mi þ rUsti þ Dt a where D/Dt is the substantial derivative following mean mixture motion (Bird et al., 2002), and ei represents the collection of terms

ei  ri

Dui þ ri ui Urvi rUri ui ui Dt

ð10Þ

i¼1

where on the right hand side, the first term represents the equilibrium stress, d being the unit tensor, or the isotropic pressure, p, while the

The quantity ei representing diffusive acceleration, convection, and stress terms, results from an effort to write the acceleration

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term in the species momentum balance equation with mass average velocity v as the frame of reference, rather than the species velocity vi, as in Eq. (7). This is done so that when the species momentum balance is summed over all n species, the mixture momentum balance of the conventional form results, as discussed below. Eq. (10) may be written in alternate forms (Bearman, 1959), e.g., by using the identity rUriuiui ¼ riuiUrui + uirUriui along with the use of the mass diffusion flux with respect to the mass-average mixture velocity, ji  riui, as well as the equation of continuity, Eq. (6)

ei ¼

Dji þ ji rUv þji UrvðMi Dri Þui Dt

ð11Þ

Only n  1 of the species momentum balance, Eq. (9), are independent (Whitaker, 1986), the nth vi being determined by the relation between mixture and species velocities, i.e., rv ¼ Pn i ¼ 1 ri vi , where v is governed by the mixture momentum balance. In fact, summing Eq. (9) over all species gives the total momentum balance of the mixture, i.e., the nth relation for the mixture, in the conventional form of a single fluid (Bird et al., ¨ 2002; Muller, 1985; Whitaker, 1986) ! n X X Dv t ð12Þ r ri f ia ¼ rp þ rUs þ Dt a i¼1 P wherein we have used the relations rp ¼ ni¼ 1 ci rT mi , as well as Pn results from summing Eq. (11), and using i ¼ 1 ei ¼ 0. The latter P in it the relation ni¼ 1 ji ¼ 0 (Bird et al., 2002), along with n X

ðMi Dri Þui ¼

q X

r¼1

i¼1

rr

n X

nri Mi vi ¼ 0

ð13Þ

i¼1

involving the assumptions of mass as well as momentum conservation in individual reactions, much like that in diffusional interactions, i.e., n X

ci Fi ¼ 0

ð14Þ

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thus, results simply from a rearrangement of Eq. (9), i.e., " # X xi Fi 1 Dv t di ¼ ði ¼ 1,2,. . .,nÞ ¼ c rT mi rUsi  ri f ia þ ri þ ei RT cRT i Dt a ð16Þ which shows it to include the chemical potential, the total partial stress, external body forces, as well as acceleration and inertial terms represented by ei. Estimates of the orders of magnitude of the different driving forces are provided by Whitaker (1986), while Bearman (1959) has commented on the importance of inertial and viscous terms for common liquids at room temperature, including those represented by ei. The term rTmi in this is sometimes written in terms of partial molar Gibbs free energy Gi , i.e., the chemical potential, and partial molar enthalpy Hi (Bird et al., 2002; Truesdell, 1962) by using T rðGi =TÞ þHi r ln T. A more common alternate form may be written with the use of the thermodynamic relation rT mi ¼ V i rp þ rT,p mi and the definition of the volume fraction of species i, i.e., fi ¼ ci V i ( ) X x F 1 Dv di ¼ i i ¼ ci rT,p mi þ fi rprUsti  ri f ia þ ri þ ei RT cRT Dt a ði ¼ 1,2,. . .,nÞ

ð17Þ

This is the form available in the literature from the continuum mechanical approach (Snell and Spangler, 1967). In order to obtain the more familiar form derived from IRT, which also has Pn  the advantage of including within it i ¼ 1 ci Fi ¼ 0, the mixture acceleration term, Dv/Dt, in Eq. (17) may be replaced by the use of the mixture momentum balance, Eq. (12), whereupon 2 xi Fi 1 4 ¼ di ¼ c rT,p mi þðfi oi ÞrpfrUsti oi rUst g RT cRT i 9 8 3 = n X< X  ri f ia oi rj f ja þ ei 5 ; a :

ð18Þ

j¼1

i¼1

also used in deriving Eq. (12). This is also consistent with momentum conservation in individual molecular encounters, i.e., Newton’s third law. The last two expressions together represent the principle that the linear momentum of a mixture (Eq. (12)) is not affected by internal momentum exchange among species (Truesdell, 1962), whether in chemical (Eq. (13)) or physical (Eq. (14)) encounters. ¨ Additionally, in Eq. (12) (Muller, 1985; Whitaker, 1986), of course

st ¼

n X

ðsi ri ui ui Þ

ð15Þ

i¼1

i.e., the total stress tensor in the mixture resulting from all shortrange interactions among the molecules. It may be noted that this tensor is symmetric (Whitaker, 1986). Kerkhof and Geboers (2005b) have alternately suggested that instead of using the mixture momentum balance, Eq. (12), as the nth relation, simply n species momentum balance equations, Eq. (9), ought to be used.

2.2. Diffusion driving force The so-called generalized ‘‘driving’’ force for diffusion di (Bird et al., 2002; Hirschfelder et al., 1964), actually, the diffusion ‘‘drag’’ force, is related to Fi by cRTdi ¼ ci Fi . An expression for this,

in which the additional terms, of course, collectively account for ri(Dv/Dt). At mechanical equilibrium, i.e., when the mixture is not accelerating, all these terms with the coefficient oi can be dropped. This is of the common form available in the literature, although the result is generally not as complete in the terms represented, and is obtained usually by considerably more obscure means (Bird et al., 2002; Cunningham and Williams, 1980; de Groot and Mazur, 1962; Hirschfelder et al., 1964; Lightfoot, 1974; Taylor and Krishna, 1993; Zhdanov et al., 1962). On the right hand side of this expression, the first term represents the so-called ordinary diffusion, the second term represents pressure diffusion, the third term is stress-gradient diffusion (Annis, 1971; Jou et al., 2001a), while the fourth term is forced diffusion. The last term, ei, is a collection of the remaining acceleration and inertial terms, often neglected except, e.g., for rapidly oscillating electric field (Fitts, 1962), or pressure. It may be noted that in the above derivation no assumptions were made as to the state of aggregation of the fluid. Thus, it is seen that a very complete expression for the diffusion driving force is obtained simply from the species linear momentum balance, which is applicable to any fluid mixture, i.e., gases, plasmas, liquids, and polymer, colloidal, aerosol, elastic, and viscoelastic mixtures. It is to be further noted that, because of P Eq. (14), i.e., ni¼ 1 di ¼ 0, only n  1 diffusion forces are independent. Furthermore, it may be recalled that the equations of conservation of mass of species and mixture, as well as that of

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linear momentum of the mixture have already been incorporated within it.

compositions, i.e., gi ¼ gi(T, p, x1, x2, y, xn  1), thus

rT,p mi ¼ RT r ln ai ¼ 2.3. Diffusion velocities and fluxes We digress here briefly to define generic diffusion velocities and fluxes, since the literature is replete with the use of various types of diffusion velocities (Bird et al., 2002; Kirkwood et al., 1960). The generic framework provided below provides flux relationships in a compact form. Thus, we define a diffusion velocity with respect to a generic b-frame of reference velocity, i.e., ubi  vi vb , where b-centered velocity of the mixture is defined by (Snell and Spangler, 1967) Pn j ¼ 1 cj bj vj ð19Þ vb  Pn j ¼ 1 c j bj For instance, the parameter bj ¼Mj yields the mass average velocity v, bj ¼1 results in the molar average velocity v*, bj ¼ V j yields the volume average velocity, v0, and bj ¼ djn provides vb ¼vn, i.e., velocity of reference species n, e.g., the solvent. The b

b

corresponding diffusion fluxes are defined by Ji  ci ui as the b

molar flux, and ji  ri ubi as the mass flux. The relation to absolute species molar flux, Ni ¼civi, is n X

Jbi ¼ Ni wbi

bj Nj

ði ¼ 1,2,. . .,nÞ

ð20Þ

j¼1

where the generic composition variable for species i, wbi , is defined by

wbi  Pn

ci

j¼1

c j bj

ði ¼ 1,2,. . .,nÞ

ð21Þ

These equations represent a general form of the various frames of reference used in the literature for defining diffusion velocity. 2.4. Constitutive equations The derivation of the diffusion driving force above involves few assumptions. In order to proceed further, however, some constitutive equations are necessary. The constitutive equations are not considered ‘‘axioms’’ in the same sense as the conservation laws (Truesdell, 1984; Whitaker, 1986), and are usually based on experimental observations and/or approximate theoretical models, e.g., Fick’s ‘‘law.’’ As such, they are often empirical, depend upon the material, and are subject to evolution. In this regard, it is to be noted that there is an extensive Mixture Theory literature on restrictions on the forms of the constitutive equations imposed by the consideration of thermodynamics, material symmetry, and material frame-indifference (Bowen, 1976). The usual form for the chemical potential, a function of temperature, pressure (equilibrium stress), and composition, e.g., is

mi ¼ moi ðT,pÞ þ RT ln ai

ð22Þ

where for the case of liquids, the activity ai ¼ gixi, while for the case of gas mixtures at low density or ideal liquids, ai ¼xi the mole fraction. An alternate constitutive equation for chemical potential involves fugacity (Tester and Modell, 1997). For polymer solutions, on the other hand, the activity is given by the Flory– Huggins equation (Flory, 1953). More generally, the chemical potential of a species may also be affected by other factors, e.g., non-equilibrium stress st, as discussed later on. For a liquid solution, since the activity coefficient itself is a function of temperature, pressure, and n  1 independent

n1 RT X G rx xi j ¼ 1 ij j

ð23Þ

where the so-called thermodynamic factor (Taylor and Krishna, 1993)   @ln gi Gij  dij þxi ð24Þ @xj T,p,x kaj

¨ where the Kronecker delta function, dij ¼0 for jai, and dij ¼1 for j¼i. For an ideal liquid or gaseous mixture, of course, gi ¼1, and Eq. (23) reduces to rT,pmi ¼(RT/xi)rxi. For the partial stress tensor of species i in the fluid mixture, sti ¼ si ri ui ui , the simplest constitutive equation is that proposed by Bearman and Kirkwood (1958), which is similar to the Newton’s equation for a single component fluid (Bird et al., 2002; Snell and Spangler, 1967), i.e.,   2 sti ¼ Zi ðrv þ rvT Þ Zi li ðrUvÞd ð25Þ 3 where Zi and li are the partial coefficients of shear and bulk P viscosity, respectively, such that for the mixture Z ¼ ni¼ 1 Zi , and Pn l ¼ i ¼ 1 li , resulting in the standard form of the phenomenological constitutive equation for the Newtonian mixture or for a single component (Bird et al., 2002)   2 st ¼ Zðrv þ rvT Þ Zl ðrUvÞd ð26Þ 3 Of course, there is no compelling reason why the constitutive equation for stress in a mixture should be of the same form as that for a pure component. Further, although the total stress tensor in a mixture is symmetric, the individual partial stress tensor sti need not be symmetric even in the absence of angular momentum supply. Additionally, recall that sti involves the diffusive stress sDi ¼ ri ui ui . In view of this, other authors have proposed alternate constitutive equations for stress in a multicomponent mixture. For example, Snell and Spangler (1967), Snell et al. (1967), and more recently Kerkhof and Geboers (2005a), have provided alternate constitutive equations for the partial stress tensor in terms of gradient of species velocity, rvi, rather than in terms of gradient of mass average velocity of mixture, rv, as in Eq. (25). Thus, the expression of Kerkhof and Geboers (2005a) for the partial stress tensor is   2 sti ¼ Zi ðrvi þ rvi T Þ Zi li ðrUvi Þd ð27Þ 3 P along with st ¼ ni¼ 1 sti for the mixture. Whether alternate expressions such as this are superior to that of Bearman and Kirkwood (1958) is open to debate, however, since relations for mixture viscosity Z in terms of pure component viscosity, Zii, are, in turn, dependent upon the form of the constitutive equation adopted (Bearman and Jones, 1960; Snell and Spangler, 1967). For obvious reasons, the question of the appropriate form of constitutive equations for the partial stress term is discussed at considerable length in the mechanical engineering literature (e.g., Atkin and Craine, 1976a; Bowen, 1976). Further, of course, the constitutive equations for partial stress in a non-Newtonian fluid, e.g., a polymer solution (Bird et al., 2002), or in a mixture of elastic solids (e.g., Bowen, 1976), are different. At any rate, any justifiable constitutive equation for partial stress in a given material may be adopted within the continuum mechanical framework. Similarly, there is controversy in the literature on the appropriate form of the constitutive equation for the internal species interaction force, Fi . In the Mixture Theory literature, in fact, it has been hypothesized to include a variety of contributions,

R. Datta, S.A. Vilekar / Chemical Engineering Science 65 (2010) 5976–5989

i.e., a drag force, Fi,D , or a frictional force, resulting from the relative velocity of species, that due to density gradients, or a buoyancy force, Fi,B , a thermophoretic force (Bielenberg and Brenner, 2005) due to temperature gradient, Fi,T , and other forces such as the Faxen’s force, the Magnus effect, and the Basset force (Bowen, 1976; Massoudi, 2003), etc., i.e., Fi ¼ Fi,D þ Fi,B þ Fi,T þ   . However, the majority of these force terms are rather esoteric and controversial. For instance, the internal buoyancy force, Fi,B , is said to be distinct (Bowen, 1976; Jackson, 2000) from the Archimidean buoyancy force which, of course, arises from the presence of the external body forces such as gravity (Bird et al., 2002). Given their speculative and controversial nature, thus, here we include only those contributions to Fi that are firmly established and are significant in diffusion in fluid mixtures, i.e., that due to relative motion, or the drag force, and that due to temperature gradient, i.e., a thermophoretic force. In other words, we assume that Fi ¼ Fi,D þ Fi,T (Lam, 2006; Ramshaw, 1993). Next, the internal force on species i as a result of its interactions with all species (other than i) may be written as Fi 

n X

cj Fij

ði ¼ 1,2,. . .,nÞ

ð28Þ

j¼1 j ai

Fij ¼ zij ðvi vj Þ þ zij

Kaper, 1972; Hirschfelder et al., 1964) and for monatomic liquids by Bearman and Kirkwood (1958). It is to be further remarked that the frictional model utilized here is valid for en masse diffusion, to which there are known exceptions. For instance, protons in aqueous solutions diffuse largely via the so-called Grotthuss diffusion mechanism (Grotthuss, 1806), whereby they simply hop from a hydronium ion to an adjacent water molecule, and so on. The rate of such diffusion is determined by the rate of rotation of the water molecule adjacent to a hydronium ion into a receptive orientation (Choi et al., 2005). In any case, the use of Eq. (29) in Eq. (28) provides ! T n n X X DTi Dj cj zij ðvi vj Þ þ cj zij  r ln T ði ¼ 1,2,. . .,nÞ Fi ¼ j¼1

DTi

ri



DTj

rj

!

r ln T

j¼1

ri

rj

ð30Þ where j ¼i has also been included in the summation, since Fii ¼0, as per Eq. (29). Further, we define the Maxwell–Stefan binary diffusion coefficient Dij by Dij 

where Fij is the interaction force (including drag and thermophoretic components) experienced per mol of species i as a result of its interaction with species j, present at unit concentration in the mixture, i.e., it is the molar analog of interaction force between a molecule each of species i and j. At the molecular level, Fij represents the momentum transfer between i and j due to molecular collisions. The hydrodynamic drag component of Fij is assumed (Bearman and Kirkwood, 1958; Laity, 1959; Lamm, 1957; Spiegler, 1958; Tyrrell and Harris, 1984) to be proportional to the relative velocity of the two species, and directed so as to oppose the motion of the species i, i.e., Fij,D ¼  zij(vi vj), where, the phenomenological frictional, or impedance, coefficient for drag, zij 40. It is similarly assumed that the thermophoretic force contribution to Fij, is proportional to the relative thermal diffusion velocity of the two species, i.e., (Bearman and Kirkwood, 1958; Lam, 2006; Ramshaw, 1993; Snell and Spangler, 1967) Fij,T ¼ zij ðDTi =ri DTj =rj Þr ln T. Here the thermal diffusion velocity is described by the phenomenological thermal diffusion flux equation (Bird et al., 2002), where DTi is the multicomponent thermal diffusion coefficient. As a result

5981

RT czij

ð31Þ

where the binary diffusion coefficient are clearly reciprocal as well, by virtue of the reciprocity of the frictional coefficients, i.e., Dij ¼ Dji. It is further evident from this relation that the quantity D0ij cDij, where c is the mixture concentration, would be expected to vary less with composition than the usual diffusion coefficient Dij (Taylor and Krishna, 1993). 2.5. Generalized Maxwell–Stefan (GMS) equations The generalized Maxwell–Stefan (GMS) equations (Lightfoot et al., 1962), so called because the original Maxwell–Stefan equations were limited to gas mixtures, result by combining Eqs. (30) and (31) and using the relation cRTdi ¼ ci Fi , i.e., (Bird et al., 2002; Curtiss, 1968; Curtiss and Bird, 1996, 1999; Cussler, 1976, 1997; Deen, 1998; Kerkhof and Geboers, 2005a; Kuiken, 1994; Lightfoot, 1974; Standart et al., 1979; Newman, 1991; Slattery, 1981, 1999; Taylor and Krishna, 1993; Tyrrell and Harris, 1984; Wesselingh and Krishna, 1990) ! T n n X X xi xj xi xj DTi Dj x F ðr lnTÞ ðvi vj Þ þ  di ¼  i i ¼ RT Dij Dij ri rj j¼1 j¼1 ði ¼ 1,2,. . .,nÞ

ð32Þ

ð29Þ

The frictional coefficients zij, in turn, are related to Spiegler’s frictional coefficient, fij, by cjzij ¼fij (Spiegler, 1958). There is an equal and opposite force experienced by species j by virtue of momentum balance, or Newton’s third law, in a molecular encounter, i.e., Fij ¼–Fji (Maxwell, 1866, 1868), which shows that zij ¼ zji, i.e., the frictional coefficients are reciprocal. The multicomponent thermal diffusion coefficients in this relation are Pn T further subject to i ¼ 1 Di ¼ 0. For gases, these transport coefficients, namely, zij and DTi , can be estimated reasonably well from elementary kinetic theory by accounting for the mean frequency of intermolecular collisions vij and the exchange of momentum in each collision Pij (Cowling, 1970; Ramshaw, 1993; Williams, 1958). Unlike zij, the multicomponent thermal diffusion coefficient DTi can be positive or negative, and is a strong function of composition. More general formal expressions for gases are given by (Chapman and Cowling, 1970; Eu, 1992; Ferziger and

which may alternately be written in terms of the diffusion velocity, ubi . ! T n n X X xi xj b xi xj DTi Dj b di ¼ ðr lnTÞ ði ¼ 1,2,. . .,nÞ ðu uj Þ þ  Dij i Dij ri rj j¼1 j¼1 ð33Þ b

b

since vi vj ¼ ui uj . An alternate definition of multicomponent thermal diffusion coefficient is sometimes used in these equations, i.e., DTij 

DTi

ri



DTj

rj

ð34Þ

Other common thermal transport coefficients used in the literature are aTij  DTij =Dij ¼ ðDTi =ri DTj =rj Þ=Dij , i.e., the multicomponent thermal diffusion factor, and the thermal diffusion ratio, kTij ¼ aTij xi xj .

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The GMS equations may also be written in the flux forms, e.g., from Eqs. (32) and (33) for the isothermal system di ¼

n n X X 1 1 ðxj Ni xi Nj Þ ¼ ðxj Jbi xi Jbj Þ cD cD ij ij j¼1 j¼1

ði ¼ 1,2,. . .,nÞ ð35Þ

and similarly for the non-isothermal case, e.g., in terms of diffusive fluxes and the thermal diffusion ratio n n X X 1 ðxj Jbi xi Jbj Þ þ xi xj aTij ðr ln TÞ di ¼ cDij j¼1 j¼1

ði ¼ 1,2,. . .,nÞ

aj Nj ¼ 0

ð37Þ

j¼1

where aj is the so-called determinancy coefficient. For example (Taylor and Krishna, 1993), when aj is a constant, this implies Pn j ¼ 1 Nj ¼ 0. On the other hand, when aj ¼ nj, the stoichiometric coefficient, the fluxes are determined by reaction stoichiometry, as for example in a catalyst. When aj ¼V j , it implies zero volumetric mixture velocity, v0 ¼0, and aj ¼ djn represents diffusion through a stagnant reference species, n. If such a boot-strap relation is not available, only relative fluxes or velocity differences can be obtained from the GMS equations. For the diffusive fluxes, the relation corresponding to Eq. (37) is (Snell and Spangler, 1967)

bj Jbj ¼ 0

ð38Þ

j¼1

which simply results from the frame of reference employed for the diffusion flux. Thus, no additional information is needed for diffusion fluxes. In other words, only n  1 diffusive fluxes are independent which can, hence, be determined uniquely from the n  1 independent GMS equations. The complete GMS relationship results by using in the above an appropriate form of the diffusion driving force. Thus, for example, Eq. (18), along with the use of Eq. (22), provides 2 di ¼ xi rlnai þ

ð41Þ

ð36Þ

It is to be noted that because of i ¼ 1 di ¼0, only n 1 GMS relations above are independent for n species absolute or diffusion velocities or fluxes. Therefore, another relation among fluxes, called the boot-strap relation by Taylor and Krishna (1993), is required in order to obtain all of the n absolute or diffusion velocities or fluxes. This relation may take the generic form for the absolute fluxes

n X

3 ) X  ðf ia f ja Þ þ ei 5 a

Pn

n X

An alternate form that is somewhat more revealing results from algebraic manipulations of the first three terms in the square brackets of Eq. (39), yielding 2 ( ! n 1 4 X 1 1 r oi oj ðui uj Þrp ðrUsti Þ ðrUstj Þ di ¼ xi rlnai þ ri rj cRT j¼1

1 6 4ðfi oi ÞrpfrUsti oi rUst g cRT

3 9 8 n = X< X  ri f ia oi rj f ja þ ei 7 5 ; a :

ð39Þ

j¼1

These are the most general form of multicomponent diffusion equations available by any means, and are applicable to a wide variety of mixtures including gases, liquids, polymeric liquids, and colloidal or aerosol mixtures. For the case of liquids, Eq. (23) may alternately be used for the first term on the right-hand side of Eq. (39), while for the case of gas mixtures at low density, the use of ai ¼xi and fi ¼xi provides 9 8 2 3 = n X X< 14 t t rpi oi rpfrUsi oi rUs g ri f ia oi rj f ja þ ei 5 di ¼ ; p a : j¼1

ð40Þ

where the partial specific volume of species i, ui ¼ V i =Mi . This new form of driving force clearly shows that pressure diffusion would occur only if the partial specific volumes of two species are different, i.e., ui a uj . Further, forced diffusion would occur only if mass specific body forces on the individual species are different, i.e., fia afja. Thus, as discussed below, gravitational and centrifugal forces cause no diffusion. Another form of the driving force may be obtained by the use of Eqs. (25) and (26) for the total and the partial stress tensor 2 ( # n X 1 4 X r oi oj ðui uj Þrp ðf ia f ja Þg di ¼ xi rln ai þ cRT a j¼1 2 ! (  n 1 4 X Zi Zj 1 r oi oj  r2 v þ rðrUvÞ  ri rj cRT 3 j¼1 ! # ) 

li

ri



lj

rj

ðrðrUvÞÞ ei

ð42Þ

Other forms of the partial stress tensor (Kerkhof and Geboers, 2005a; Snell and Spangler, 1967) may, of course, alternately be used in Eq. (41). Thus, the complete GMS diffusion equations result, e.g., from Eqs. (32), (34), and (41) n X xi xj

n X xi xj

DTij ðr ln TÞ D ij j¼1 j¼1 2 ( ! n 1 4 X 1 1 t t r oi oj ðui uj Þrp ðrUsi Þ ðrUsj Þ ¼ xi r ln ai  ri rj cRT j¼1 Dij

ðvi vj Þ þ

9 3 = X  ðf ia f ja Þ þ ei 5 ; a

ð43Þ

or, in alternate form, from Eqs. (32) and (39) ! T n n X X xi xj xi xj DTi Dj ðrlnTÞ ðv v Þ þ  Dij i j Dij ri rj j¼1 j¼1 2   1 4 ¼ xi rln ai  ðfi oi Þrp rUsti oi rUst cRT 9 8 3 n = X X< ri f ia oi rj f ja þ ei 5  ; a :

ð44Þ

j¼1

which for the isothermal case reduces to 2 n X   xi xj 1 4 ðvi vj Þ ¼ xi r ln ai  ðfi oi Þrp rUsti oi rUst D cRT ij j¼1 9 8 3 = n X X< ri f ia oi rj f ja þ ei 5  ð45Þ ; a : j¼1

R. Datta, S.A. Vilekar / Chemical Engineering Science 65 (2010) 5976–5989

Other forms may, of course, be written, e.g., by using Eq. (17) instead for di in Eq. (32) ! T n n X X xi xj xi xj DTi Dj ðr ln TÞ ðvi vj Þ þ  Dij Dij ri rj j¼1 j¼1 ( ) X 1 Dv t ð46Þ ¼ ci rT,p mi þ fi rprUsi  ri f ia þ ri þ ei cRT Dt a The GMS equations above have been written in terms of absolute species velocity difference. Alternately, of course, they may be written in terms of diffusion velocities (cf. Eq. (33)) or in terms of absolute or diffusive fluxes (cf. Eq. (35)). 2.6. Generalized Fick–Onsager (GFO) equations for isothermal case While the GMS form of flux relations, in which the diffusion driving force for the ith species di is written as a linear combination of the fluxes of all species, are convenient for many computations (Kaza and Jackson, 1980; Skrzypek et al., 1984), in other cases it is desirable to invert them so as to obtain the flux of a species i as a linear combination of the diffusion driving forces dj for all species, i.e., the GFO form (Datta et al., 1992; Thampan et al., 2000). Owing to the assumed linearity between flux and driving forces, and assuming no velocity dependence in di, the GMS equations involving linear driving forces can be formally inverted through the use of linear algebra (Bearman, 1958, 1959; Bearman and Kirkwood, 1958; Fitts, 1962; Krishna, 1987b; Snell and Spangler, 1967), although the explicit expressions become unwieldy for n Z3 (Amundson et al., 2003). We start, for the isothermal case, by writing Eq. (35) for the diffusive flux in the form cdi ¼

n X

Fij Jbj

ði ¼ 1,2,. . .,nÞ

ð47Þ

j¼1

where the effective GMS frictional coefficients 1 0 n BX xk C xi C B Fij  dij B C þ ðdij 1Þ A @ D D ij ik k¼1

ð48Þ

ka i

5983

where the matrix [Fb] of elements Fijb is a square matrix of size (n 1)  (n 1) that is nonsingular, and hence may be inverted to formally provide explicit GFO relations for the diffusive flux ðJb Þ ¼ c½Gb ðdÞ

ð53Þ

The expression may alternately be written as Jbi ¼ c

n1 X

Gbij dj

ði ¼ 1,2,. . .,n1Þ

ð54Þ

j¼1

which is of the GFO form. Here ½Gb  ¼ ½F b 1 , i.e., elements Gbij of the matrix [Gb] are those of the inverse matrix ½F b 1 , and may, hence, be called effective GFO diffusion coefficients. With the aid of Cramer’s rule and certain properties of determinants (Strang, 1988), Eq. (54) may alternately be written in the form   b b  Fb F21 ::: Fðn1Þ1 0   11     b b b  F12 F22 ::: Fðn1Þ2 0    ^ ^ ::: ^ ^     b b b  F  1ði1Þ F2ði1Þ ::: Fðn1Þði1Þ 0   c  b b b b Ji ¼  F1i F2i ::: Fðn1Þi 1  ði ¼ 1,2,. . .,n1Þ ð55Þ D    b b b  F1ði þ 1Þ F2ði þ 1Þ ::: Fðn1Þði þ 1Þ 0     ^ ^ ::: ^ ^     b b b F F1ðn1Þ 0   1ðn1Þ F1ðn2Þ :::    d1 d2 ::: dn1 0 where D is the determinant of the matrix [Fb], i.e.,   b b b   F F12 ::: F1ðn1Þ   11   b b   Fb F22 ::: F2ðn1Þ   21 D¼    ^ ^ ::: ^     b b b  Fðn1Þ1 Fðn1Þ2 ::: Fðn1Þðn1Þ 

ð56Þ

Similarly, the explicit form for absolute flux may be obtained by combining Eqs. (35) and (37) and solving as above for flux Ni ¼ c

n1 X

Gaij dj

ði ¼ 1,2,. . .,n1Þ

ð57Þ

j¼1

The GMS effective frictional coefficients defined thus are without the benefit of reciprocity, but that is of little concern here. The above equations may be written in matrix form for inversion, but the resulting matrix [F] of the effective frictional coefficients Fij is singular. Thus, one of the fluxes, say Jbn , must be eliminated from Eq. (47). For this, we use Eq. (38), i.e., Jbn ¼ 

n1 1 X

bn j ¼ 1

bj Jbj

ð49Þ

where effective diffusional coefficients Gaij are the elements of the matrix ½Ga  ¼ ½F a 1 , with elements of the effective frictional coefficient matrix ½F a 1 being given by 1 0   n BX xk a x C x a x C B ð58Þ Fija  dij B þ i i C þ ðdij 1Þ i  i i @ D an Din A Dij an Din k ¼ 1 ik k ai

Eq. (57) may also be written in a form similar to Eq. (55).

in Eq. (47), resulting in cdi ¼

n1 X

Fijb Jbj

ði ¼ 1,2,. . .,n1Þ

ð50Þ

3. Discussion and limiting cases

ð51Þ

Even though our purpose here is to illuminate the continuum mechanical approach to the general multicomponent diffusion equations, not discuss their various applications, nonetheless, a few examples and limiting cases are presented.

j¼1

where 1

0

  n BX xk b x C x b x C B Fijb  dij B þ i i C þ ðdij 1Þ i  i i @ D bn Din A Dij bn Din k ¼ 1 ik

3.1. Body forces

k ai

Eq. (50) may be written in the matrix form b

b

cðdÞ ¼ ½F ðJ Þ

ð52Þ

The body forces normally considered are gravitational, centrifugal force, and that due to the electric field E on a charged particle (Horne, 1966). However, there clearly are other body

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R. Datta, S.A. Vilekar / Chemical Engineering Science 65 (2010) 5976–5989

forces that may be significant in special cases. For instance, Bird et al., (2002) provide the example of force on a charged particle moving in an electric and a magnetic field, including the forces of electrical and magnetic induction resulting from Lorentz relation. When these are included, the various body forces acting on a charged species i X   z   f ia ¼ g þ o2 r þ2ðvi  xÞ þ i F E þ ðvi  BÞ Mi a o 1 n el þ wi ðEUrEÞ þ wmag ðBUrBÞ ð59Þ i Mi mag are the electric where B is the magnetic induction and wel i and wi and magnetic susceptibility of species i, respectively. Such terms are significant, e.g., in mass spectrometry, plasma dynamics (Chen, 1984; Zhdanov, 2002), dielectrophoresis, and magnetophoresis. Most of these terms are seen to be nonlinear and, thus, are not strictly compatible with the linear IRT formalism (Yao, 1981). However, there is no such issue in including the various driving forces within the framework of the continuum mechanical approach, although the inversion to obtain the GFO form is not possible when terms containing species velocity are present in the driving force. Eq. (59) also contains the Coriolis force, 2vi  x, although it is usually negligible compared with the centrifugal force o2r (Yao 1981). Not all of these forces, of course, directly cause forced diffusion (Eq. (41)), although they can affect mixture flow and, hence, stress as well as pressure gradients (Eq. (12)). In short, all conceivable body forces may be incorporated within the continuum mechanical formalism of multicomponent diffusion.

3.2. Mechanical equilibrium The expressions for di given above are very general, and not all terms are always significant. In fact, substantial simplifications are warranted under most practical situations. For instance, under mechanical steady-state, i.e., when the species i is not accelerating, ai ¼0 (Eq. (7)). If we further assume that ei E0, then from Eq. (44) ! T n n X X xi xj xi xj DTi Dj ðr ln TÞ ðv v Þ þ  Dij i j Dij ri rj j¼1 j¼1 2 1 4 ¼ xi r ln a ðfi oi ÞrpfrUsti oi rUst g cRT 93 8 n = X X< ð60Þ ri f ia oi rj f ja 5  ; a : j¼1

Order of magnitude analysis of the terms in the diffusion equation by Whitaker (1986) and Bearman (1959) show that the effect of species acceleration is almost always negligible for both gases and liquids, except for example in a rapidly oscillating electric field (Fitts, 1962). It may be further argued that for the case of mechanical equilibrium, since for ri(Dv/Dt)¼0 as well, the additional terms in the driving force Eq. (18), as a result of using Eq.(12), should also be dropped. This provides ! T n n X X xi xj xi xj DTi Dj ðrln TÞ ðv v Þ þ  Dij i j Dij ri rj j¼1 j¼1 " # X 1 t ð61Þ fi rprUsi  ri f ia ¼ xi rlnai  cRT a which could alternately be obtained directly from Eq. (46). On the other hand, however, the additional terms may equally well be

retained, as is the common practice, since under these conditions they together add up to zero from the equation of motion for mixture, Eq. (12), i.e., ! n X X ð62Þ r p ¼ r Us t þ ri f ia a

i¼1

When, in addition to zero acceleration, the velocity gradients and hence the stress tensor vanish (Bearman, 1959; Fitts, 1962) for the case of mechanical equilibrium, as for example in a centrifuge wherein the fluid undergoes rigid body rotation, then from Eq. (60) ! T n n X X xi xj xi xj DTi Dj ðr lnTÞ ðvi vj Þ þ  Dij Dij ri rj j¼1 j¼1 93 8 2 = n X X< 1 4 ri f ia oi rj f ja 5 ð63Þ ðfi oi Þrp ¼ xi r ln ai  ; cRT a : j¼1

which is the most common form of expression available for multicomponent diffusion as derived from IRT (Bird et al., 2002; Merk, 1959). Under these conditions, of course, from Eq. (62) ! n X X ð64Þ rp ¼ r i f ia a

i¼1

3.3. Gases and plasmas For the case of a gaseous mixture, Eq. (60), with the use Eqs. (25) and (26) and the assumption of zero bulk viscosity for gases, reduces to n X xi xj j¼1

Dij

ðvi vj Þ þ

n X

xi xj aTij ðr ln TÞ

j¼1

  1 1 rpi oi rpðZi oi ZÞ r2 v þ rðrUvÞ p 3 93 8 = n X X< ri f ia oi rj f ja 5  ; a :

¼

ð65Þ

j¼1

which is of the form provided by the Grad–Zhdanov theory (Cunningham and Williams, 1980; Mason et al., 1967; Zhdanov et al., 1962). In the absence of the stress terms, i.e., for mechanical equilibrium ! T n n X X xi xj xi xj DTi Dj ðr ln TÞ ðvi vj Þ þ  Dij Dij ri rj j¼1 j¼1 93 8 2 n = X< X 14 ð66Þ ¼ rpi oi rp ri f ia oi rj f ja 5 ; p a : j¼1

which is the expression resulting from the conventional Chapman– Cowling kinetic theory (Hirschfelder et al., 1964). Higher Chapman–Cowling approximations to the distribution function are necessary to predict the stress induced diffusion (Annis, 1971). For the isothermal case, and further making use of Eq. (64) (Truesdell, 1962) in this ! n X X xi xj 1 ð67Þ ðvi vj Þ ¼  rpi  ri f ia Dij p a j¼1 These equations are also applicable to plasmas (Chen, 1984; Zhdanov, 2002). For instance, using Eq. (59) in Eq. (67) for the electric and magnetic forces, and assuming that the electric and

R. Datta, S.A. Vilekar / Chemical Engineering Science 65 (2010) 5976–5989

magnetic susceptibilities of plasma components are zero, provides (Chen, 1984) n X xi xj j¼1

ðvi vj Þ ¼ 

Dij

  1

rpi ci zi F E þ ðvi  BÞ p

ð68Þ

Further, in the absence of external fields, the results reduce to the classical Maxwell–Stefan equations (Whitaker, 1986) n X pi pj j¼1

pDij

ðvi vj Þ ¼ rpi

J1 ¼ 

D12 ðc1 rT,p m1 rUst1 Þ RT

For the case of an electric field, with E¼  rf, the external body force on charged species i is fia ¼  ziFrf/Mi. Thus, ðri f ia oi Pn Pn Pn j ¼ 1 rj f ja Þ ¼ zi ci F rf, since j ¼ 1 rj f ja ¼ F r f j ¼ 1 zj cj ¼ 0 Pn because of local electroneutrality condition, i ¼ 1 zi ci ¼ 0, which is almost universally applicable except in the neighborhood of charged surfaces (Newman, 1991). However, in general, rpa0 if other body forces are present (Eq. (64)). Thus, the GMS equations for electrolyte solutions (Newman, 1967, 1991) ! T n n X X xi xj xi xj DTi Dj ðr lnTÞ ðv v Þ þ  Dij i j Dij ri rj j¼1 j¼1  1  c rT,p mi þ ðfi oi Þrp þ zi ci F rf cRT i

ð70Þ

ð73Þ

which clearly shows the coupling between the divergence of the partial stress tensor and the polymer diffusion flux. Here st1 is the partial non-equilibrium stress due to the polymer. In the case of an ideal solution, this reduces to the relation of Apostolakis et al. (2002) D12 rUst1 RT

J1 ¼ D12 rc1 þ

3.4. Electrolyte solutions

¼

equilibrium. Then, from Eqs. (35) and (17), the polymer diffusion flux, with respect to the mixture molar average velocity v* (bj ¼1),

ð69Þ

where the product pDij is a constant for gases (Eq. (31)), since Dij is inversely proportional to pressure p.

5985

ð74Þ

from which it is clear that a non-uniform stress will result in a diffusive flux, or in a steady-state polymer concentration gradient within a closed system, when its diffusion flux is zero, as, e.g., at steady-state in a rotational viscometer. This is indeed found to be the case (Agarwal et al., 1994). Further, while the concentration gradient is smooth at low rotation rates, it is found that when rotation rate, e.g., in a plateand-cone rheometer, increases beyond a critical value, the polymer solution separates into distinct bands. Thus, Jou et al. (2001a) have attempted to explain this bifurcation based on the fact that, in general, the chemical potential (cf. Eq. (22)) of a polymer in solution in Eq. (73) is not only affected by its composition, e.g., as described by the Flory–Huggins equation (Flory 1953), but also by the non-equilibrium stress st1 , which, furthermore can affect the polymer solution viscosity (i.e., shear thinning, or shear thickening). A complete quantitative explanation of stress gradient induced diffusion of polymers is still elusive, but this is an active area of current research. This phenomenon is not only of theoretical significance, but it has practical implications in chromatography and separation, for instance, of proteins and DNA, etc.

which for the isothermal case may be written in the form (Krishna, 1987a) n X  1 1  ðxj Ni xi Nj Þ ¼  ci rT,p mi þ ðfi oi Þrp þ zi ci F rf cD cRT ij j¼1

ð71Þ

For the case of dilute solutions, with xn-1 for the solvent species n, and xj-0 for all other species, and in the absence of other body forces and thus rp ¼0 (Eq. (64)), this reduces to the Nernst–Planck equation (Krishna, 1987b; Newman, 1991; Taylor and Krishna, 1993) Ni ¼ 

Din ðc rT,p mi þ zi ci F rfÞ þci vn RT i

ði ¼ 1,2,. . .,n1Þ

ð72Þ

The current density, of course, may be obtained from the P relation, i ¼ F ni¼ 1 zi Ni . 3.5. Stress gradient-induced diffusion of polymers Even though the classical thermodynamics of irreversible processes does not permit a flux–force relation between diffusion and viscous stress, this phenomenon has been experimentally observed in polymer solutions for some time (Agarwal et al., 1994), and its explanation is, consequently, now included in ‘‘extended’’ irreversible thermodynamic (Jou et al., 2001b), as well as in more recent kinetic theory (Curtiss and Bird, 1996) formulations of diffusion equations. To illustrate stress gradient-induced diffusion, we consider the case of an isothermal binary mixture of a polymer ‘‘1’’ in a solvent ‘‘2’’ in the absence of external forces as well as any pressure gradient, but with molecular (viscous) stresses present, as, e.g., in a Couette, or a plate-and-cone, viscometer, under mechanical

3.6. Sedimentation and centrifugation Let us consider the rather general case of sedimentation in the presence of gravitational, centrifugal, and electric fields (Deen, 1998; Hsu, 1981; Yao, 1981). In centrifugation, the Coriolis force is usually negligible as compared to the centrifugal force, i.e., 2vi  x 5 o2 r. The stress term is also zero due to rigid body rotation (Bird et al., 2002). It is further clear from Eq. (41) that gravity and centrifugal forces cause no diffusional segregation. However, these forces do result in a pressure gradient, as per Eq. (64), which for the present case is

rp ¼ rgþ ro2 r

ð75Þ

and can, hence, result in segregation. In the above, the electric field term disappears again by virtue of electroneutrality. Using this expression in Eq. (43) to replace the pressure gradient in this case of mechanical equilibrium n X xi xj j¼1

Dij

ðvi vj Þ 2

¼ xi r lnai 





n z r 4X z oi oj ðui uj Þðrg þ ro2 rÞ þ F rf i  j

cRT

j¼1

Mi

Mj

3  5

ð76Þ The centrifugal separation is, thus, caused by the strong pressure gradient generated by the centrifugal force (Deen, 1998) rather than by the centrifugal force itself. This equation also provides the sedimentation potential (de Groot and Mazur, 1962; Yao, 1981).

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3.7. Generalized Fick’s Law for binary mixture For the case of a binary mixture of species 1 and 2, from Eqs. (36) and (38)   b2 d þx x aT rlnT ð77Þ Jb1 ¼ cD12 b1 x1 þ b2 x2 1 1 2 12 where the general driving force from Eq. (41) is ( "   @ln g1 1 rx1 þ ro1 o2 ðu1 u2 Þrp d1 ¼ 1þ @ln x1 T,p cRT # )  t t X rUs1 rUs2  ðf 1a f 2a Þ þ e1  

r1

r2

ð78Þ

a

This is a generalization of Fick’s law to include generic frames of reference and driving forces besides simply composition gradient, and is the most complete form available in the literature. When e1 and stress terms are dropped these equations yield the expression of Bird et al. (2002) for flux with respect to molar velocity of mixture (bj ¼ 1). 3.8. GFO equations for the ternary mixture For the ternary case of a mixture composed of species 1, 2, and 3, Eqs. (55) and (56) yield for the two independent diffusive fluxes with respect to mixture molar average velocity v* (bj ¼1)      F11 F21 1   c c      F F22 0  ¼    J1 ¼      12  F  Þ ðF22 d1 F12 d2 Þ ðF11 F22 F12 21  F11 F12   d1 d2 0        F21 F22 ð79Þ and J2 ¼    F11    F21

c

   F11   F     12 F12   d1   F22 

 F21  F22

d2

 0  c   1  ¼     F  Þ ðF11 d2 F21 d1 Þ  ðF11 F22 F12 21 0

multicomponent fluid of arbitrary state of aggregation, and without the approximations inherent in the conventional approaches of irreversible thermodynamics, kinetic theory of gases, or statistical mechanical theory of liquids. Further, with the adoption of appropriate phenomenological constitutive equations for species activity, that based on the frictional hypothesis for en masse diffusion, thermal diffusion, and the generalized Newtonian equation for viscous stress, a comprehensive theory of the generalized Maxwell–Stefan form results. This may be inverted to obtain the generalized Fick–Onsager form. The results are valid for any type of fluid, e.g., liquids, gases, plasmas, polymer and colloidal solutions and aerosols. Further, they form the starting point for theories of transport in porous media and in membranes, either via the dusty-fluid approach or via volume averaging. A new form of the diffusion driving force provided here clearly shows that diffusional segregation due to pressure gradient will occur only if the partial specific volumes of two species are different, while forced diffusion would occur only if mass specific body force on the individual species is different. It is hoped that the clear and comprehensive exposition of continuum mechanical diffusion theory provided here that unifies previous work, and is without the conceptual burden associated with the traditional irreversible thermodynamics approach and the tedium of the statistical theories of molecular dynamics, will take root in chemical engineering curricula and motivate further applications of the theory.

Nomenclature

Fij

activity of species i acceleration of constituent i, m/s2 magnetic induction, kg/s2A mixture molar concentration, kmol/m3 concentration of species i, kmol/m3 diffusion ‘‘driving’’ force of constituent i, xi Fi =RT, m  1 Maxwell–Stefan mutual binary diffusion coefficient for species i and j, m2/s diffusion coefficient for interaction of species i and matrix M, m2/s multicomponent thermal diffusion coefficient, kg/m s multicomponent thermal diffusion coefficient, (DTi =ri DTj =rj ), m2/s the product cDij, kmol/m s electric field, rf, V/m Faraday constant, 96,487 C/mol body force of type a per unit mass of species i, N/kg i Spiegler’s friction coefficient frictional force experienced per mol of species i via its interaction with species j, present at unit concentration effective GMS frictional coefficients, Eq. (48), s/m2

Fijb

effective GMS frictional coefficients for diffusional flux,

ai ai B c ci di Dij DiM

ð80Þ where from Eq. (51), the effective frictional coefficients are     x1 x2 x3 1 1   F11 , F12 , ¼ þ þ ¼ x1  D13 D12 D13 D12 D13     1 1 x1 x2 x3   , F22 ð81Þ F21 ¼ x2  ¼ þ þ D12 D23 D23 D12 D23

When these are used in Eqs. (79) and (80), the expressions provided by Taylor and Krishna (1993) for the case of ternary gas   mixture result. It may be noted that F12 a F21 . Similarly, results for systems with greater number of components may be obtained (e.g., Amundson et al., 2003). Further, in a similar manner, one can obtain expressions for the absolute species flux N1 and N2 for the ternary case from Eqs. (57) and (58), as done, e.g., by Pollard and Newman (1979) for a molten-salt electrolyte.

4. Conclusions

DTi DTij D0ij E F fia fij Fij

Fa ij

[Fb] [Fa] Fi Fi,D

This article explicates and extends the straightforward approach of continuum mechanics to obtain a remarkably complete description of multicomponent diffusion involving driving forces of all manner simply from the application of the principle of linear momentum balance to a species in a

Fi,B Fi,T

Eq. (51), s/m2 effective GMS frictional coefficients for absolute flux, Eq. (58), s/m2 matrix with elements Fijb matrix with elements Fija the force of interaction exerted on each mol of i by all other constituents, N/kmol i drag force, or friction, exerted on species i by other constituents, N/kmol i buoyancy force acting on i due to density gradients, N/kmol i thermophoretic force acting on i due to temperature gradients, N/kmol i

R. Datta, S.A. Vilekar / Chemical Engineering Science 65 (2010) 5976–5989

[Gb] Gbij [Ga] Gaij Gi Hi i Ji ji Jbi b

ji

kB kTij Mi n n Ni p pi Pij q r rr R Si(t) t T ui ubi v vb v0 v* vi Vi(t) ui Vi xi zi

matrix with elements Gbij ¼ [Fb]  1 elements of matrix [Gb], effective GFO diffusion coefficients for diffusional flux, m2/s matrix with elements Gaij ¼[Fa]  1 elements of matrix [Ga], effective GFO diffusion coefficients for absolute flux, m2/s partial molar Gibbs free energy of species i partial molar enthalpy of species i current density, A/cm2 diffusive molar flux of species i with respect to mass average velocity ciui, kmol/m2 s diffusive mass flux of species i with respect to mass average velocity riui, kg/m2 s diffusion molar flux of species i with respect to b-centered velocity of the mixture, ci ubi , kmol/m2 s diffusion mass flux of species i with respect to b-centered velocity of the mixture, ri ubi , kg/m2 s Boltzmann constant thermal diffusion ratio, aTij xi xj molar mass of species i, kg/kmol unit normal vector to surface total number of species absolute flux of species i, civi , kmol/m2 s pressure, N/m2 partial pressure of species i, N/m2 momentum exchange in intermolecular collision between species i and j number of chemical reactions in the system position vector, m molar rate of reaction r, kmol/m3 s universal gas constant, J/kmol K bounding surface of control volume, m2 time, s absolute temperature, K diffusion velocity of species i with respect to mass average velocity, vi  v, m/s diffusion velocity of species i with respect to b-centered velocity of mixture,vi  vb, m/s mass-average convective velocity of mixture, m/s b-centered continuum velocity of the mixture, m/s, Eq. (19) volume average velocity of the mixture, m/s molar average velocity of the mixture, m/s absolute velocity of species i, m/s control volume, m3 partial specific volume of species i, V i =Mi , m3/kg partial molar volume of species i, m3/kmol mole fraction of species i, ci/c charge number of species i

Greek letters

aj aTij bj

gi gij Gij dij d Dri

ei Z Zi

the determinancy coefficient, Eq. (37) thermal diffusion factor, DTij =Dij ¼ ðDTi =ri DTj =rj Þ=Dij generic frame of reference for diffusion, Eq. (19) activity coefficient thermphoresis drag coefficient, (N/m3)(m3/kmol)2 thermodynamic factor, Eq. (24) ¨ Kronecker delta function, dij ¼ 0 for j ai, and dij ¼1 for j¼i unit tensor net molar rate of production of species i from chemical reactions, kmol/m3 s collection of acceleration and inertial terms, Eqs. (10) or (11) fluid mixture viscosity coefficient, N s/m2 partial coefficient of shear viscosity, N s/m2

Zii l li

mi nri vij

r r ri p pi s si sDi sti st fi f

o oi wai weli wmag i zij

5987

pure component shear viscosity, N. s/m2 fluid mixture coefficient of bulk viscosity partial coefficient of bulk viscosity chemical potential of species i, J/kmol the stoichiometric coefficient of species i in reaction r intermolecular collision frequency between species i and j generic reaction density, kg/m3 partial mass density of species i, kg/m3 molecular stress tensor of mixture, N/m2 partial molecular stress tensor associated with component i, N/m2 non-equilibrium (viscous) molecular stress tensor of mixture, N/m2 non-equilibrium (viscous) partial molecular stress tensor of component i, N/m2 partial diffusive molecular stress tensor, riuiui, N/m2 2 total partial stress tensor of species i, ¼ si + sD i , N/m 2 total stress tensor of mixture, N/m volume fraction of species i, ciV i electric potential, V angular velocity of rotation mass fraction of species i, ri/r generic composition variable of species i, Eq. (21) the electric susceptibility of species i, the electric susceptibility of species i, frictional, or impedance, coefficient for interaction between species i and j, (J.s/m5)(m3/kmol)2

Subscripts i, j T, p

a

species i, j at constant temperature T, p type of body force

Superscripts a e o T

with respect to a-frame of reference effective standard state transpose; thermal

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